from sage.modular.dirichlet import DirichletCharacter
H = DirichletGroup(5635, base_ring=CyclotomicField(44))
M = H._module
chi = DirichletCharacter(H, M([33,0,6]))
pari: [g,chi] = znchar(Mod(148,5635))
Basic properties
Modulus: | \(5635\) | |
Conductor: | \(115\) | sage: chi.conductor()
pari: znconreyconductor(g,chi)
|
Order: | \(44\) | sage: chi.multiplicative_order()
pari: charorder(g,chi)
|
Real: | no | |
Primitive: | no, induced from \(\chi_{115}(33,\cdot)\) | sage: chi.is_primitive()
pari: #znconreyconductor(g,chi)==1
|
Minimal: | yes | |
Parity: | even | sage: chi.is_odd()
pari: zncharisodd(g,chi)
|
Galois orbit 5635.cc
\(\chi_{5635}(148,\cdot)\) \(\chi_{5635}(442,\cdot)\) \(\chi_{5635}(638,\cdot)\) \(\chi_{5635}(687,\cdot)\) \(\chi_{5635}(1422,\cdot)\) \(\chi_{5635}(1667,\cdot)\) \(\chi_{5635}(2108,\cdot)\) \(\chi_{5635}(2353,\cdot)\) \(\chi_{5635}(2402,\cdot)\) \(\chi_{5635}(2843,\cdot)\) \(\chi_{5635}(2892,\cdot)\) \(\chi_{5635}(3333,\cdot)\) \(\chi_{5635}(3823,\cdot)\) \(\chi_{5635}(4068,\cdot)\) \(\chi_{5635}(4362,\cdot)\) \(\chi_{5635}(4607,\cdot)\) \(\chi_{5635}(4803,\cdot)\) \(\chi_{5635}(5048,\cdot)\) \(\chi_{5635}(5097,\cdot)\) \(\chi_{5635}(5587,\cdot)\)
sage: chi.galois_orbit()
order = charorder(g,chi)
[ charpow(g,chi, k % order) | k <-[1..order-1], gcd(k,order)==1 ]
Related number fields
Field of values: | \(\Q(\zeta_{44})\) |
Fixed field: | \(\Q(\zeta_{115})^+\) |
Values on generators
\((3382,346,2696)\) → \((-i,1,e\left(\frac{3}{22}\right))\)
First values
\(a\) | \(-1\) | \(1\) | \(2\) | \(3\) | \(4\) | \(6\) | \(8\) | \(9\) | \(11\) | \(12\) | \(13\) | \(16\) |
\( \chi_{ 5635 }(148, a) \) | \(1\) | \(1\) | \(e\left(\frac{1}{44}\right)\) | \(e\left(\frac{19}{44}\right)\) | \(e\left(\frac{1}{22}\right)\) | \(e\left(\frac{5}{11}\right)\) | \(e\left(\frac{3}{44}\right)\) | \(e\left(\frac{19}{22}\right)\) | \(e\left(\frac{5}{22}\right)\) | \(e\left(\frac{21}{44}\right)\) | \(e\left(\frac{7}{44}\right)\) | \(e\left(\frac{1}{11}\right)\) |
sage: chi.jacobi_sum(n)